Fourier Analytic Approach to Quantum Estimation of Group Action

Abstract

This article proposes a unified method to estimation of group action by using the inverse Fourier transform of the input state. The method provides optimal estimation for commutative and non-commutative group with/without energy constraint. The proposed method can be applied to projective representations of non-compact groups as well as of compact groups. This paper addresses the optimal estimation of R, U(1), SU(2), SO(3), and R^2 with Heisenberg representation under a suitable energy constraint.

Summary of obtained results with energy constraints were added

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• A protocol for global multiphase estimation
• Indefinite causal order strategy does not improve the estimation of group action
• Heisenberg scaling based on population coding
• The Cramér-Rao approach and global quantum estimation of bosonic states